Many of the papers cited up to this point propose probability distributions which can be seen as alternative posteriors, in the sense that they offer quantification of uncertainty. [1]} call these alternative data-informed mappings “belief updates" and characterise the optimal, valid and order-coherent belief update as a Gibbs posterior in a Generalised Bayes setting. Several existing belief updates, such as the power posterior, are known to be valid. However the characterisation of a valid belief update holds for losses which are additive. We extend the class of losses for which valid belief updates can be defined. In particular we show the Cut-models and \(\eta \) -SMI have a kind of prequential additivity which is sufficient for the theory of [1]} to apply. We point out alternative SMI procedures, which we call \(\delta \) -SMI and \(\gamma \) -SMI. These offer different interpolating sequences of candidate posterior distributions. The \(\delta \) -SMI sequence progressively “blurs" the data away using a procedure resembling Approximate Bayesian Computation (ABC) but otherwise offers candidate posterior distributions which are often similar to those of \(\eta \) -SMI. [3]} consider a parallel framework for deriving belief updates due to [4]}. This resembles PAC-Bayesian approaches [5]}, [6]} and is distinct from the approach of [1]}.
